Title Page (include title; complete author contact information and affiliations; and acknowledgments)
This is the manuscript draft as submitted to Forest Science first. The accepted version Manso, R., Ningre, F. and Fortin, M. 2018. Simultaneous prediction of plot-level and tree-level harvest occurrences with correlated random effects. Forest Science. DOI: 10.1093/forsci/fxy015. will be available on-line at the journal site in due course.
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Simultaneous prediction of plot-level and tree-level harvest occurrences with correlated random effects Rub´en Manso1,2,* , Franc¸ois Ningre1 , and Mathieu Fortin3
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1 INRA,
UMR 1092 LERFoB, 1 rue de l’Arboretum, 54280, Champenoux, France
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2 Forest
Research, Northern Research Station, Roslin, Midlothian, UK, EH25 9SY
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3 AgroParisTech,
UMR 1092 LERFoB, 14 rue Girardet, 54042 Nancy, France
* Corresponding
author: e-mail:
[email protected],
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[email protected]
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ACKNOWLEDGEMENTS
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The authors want to thank all the people involved in the measurements of the permanent-
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plot network and specially to Daniel Ritti´e (LERFoB) who gathered and formatted the data.
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Special thanks are due to Adam Ash (Forest Research) who thoroughly proofread the original
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version of the manuscript. The UMR 1092 LERFoB is supported by a grant overseen by the
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French National Research Agency (ANR) as part of the “Investissements d’Avenir” program
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(ANR-11-LABX-0002-01, Lab of Excellence ARBRE).
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Management and Policy Implications
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Management and Policy Implications
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The probability that a plot is thinned and the probability that a tree within that plot
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is harvested are likely to be correlated. Neglecting this correlation would lead to an
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underestimation of the variance of prediction error in harvest models. As a result, forest
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managers using growth simulators that include two-level harvest models would have a
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false idea of precision in stand projections. This may result in wrong decisions in the
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long run. The proposed method deals with this issue and provides theoretically unbiased
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variance estimates.
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Manuscript (Text only; do not embed figures or tables)
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Simultaneous prediction of plot-level and tree-level harvest occurrences with correlated random effects
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ABSTRACT
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In forestry, harvest models have become popular for forecasting thinning under business-as-
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usual scenarios. There are two binary processes involved in thinning operations: (i) whether a
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plot is to be thinned and (ii) whether a particular tree within that plot is to be harvested. These
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processes can be modeled using logistic regressions. The data used to fit such models come
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from forest inventories, where the observations are not usually independent. Random effects
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can be used to deal with these correlations. However, fitting the plot-level and tree-level models
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independently hinders the estimation of the covariance between the random effects of both
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models. The objective of this paper was to develop a statistical method for the simultaneous
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prediction of harvest probabilities at the plot and tree levels in a single mixed-effects model.
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We developed a maximum likelihood estimator based on the joint distribution of the probability
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that a given plot is thinned and the probability that a given tree within that plot is harvested.
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The estimator was derived from a zero-altered binomial form, but it assumed distinct harvest
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probabilities for each single tree. The estimator was tested in the case study of mixed stands of
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oak (Quercus spp.) and beech (Fagus sylvatica L.) in Northern France.
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Keywords: harvest models, zero-altered models, mixed-effects models, simultaneous esti-
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mation, joint distribution
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INTRODUCTION
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Forest dynamics simulators require at least a growth module and a mortality module, which
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are usually implementations of statistical models. When these simulators are intended to make
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predictions of forest evolution in managed stands, a harvest module is also needed (Fortin 2014).
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In contrast to growth and mortality, harvest modules have traditionally relied on algorithms that
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deterministically decide when, where and how to cut according to some management rules (e.g.
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Pukkala and Miina 1998). The use of these algorithms is conceptually straightforward if we
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assumed that the realized harvesting strictly follows what was planned. However, for many
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reasons, the intended management often differs from its practical application. When working
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with business-as-usual scenarios statistical harvest models aimed at emulating real management
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decisions are a competing alternative to traditional algorithms (Antón-Fernández and Astrup
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2012).
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Technically speaking, harvest models have to deal with a binary outcome at both the plot
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and the tree levels (i.e. either a plot is thinned or not; either a tree is cut down or not). Lo-
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gistic regression is the usual statistical technique to model such response variables (cf. Hosmer
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and Lemeshow 2000). Therefore, the output of harvest models is the probability that a plot be
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thinned or that a tree be harvested, depending on the level at which the model applies. Among
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the few papers on this subject, several studies have strictly focused on forecasting the proba-
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bilities at the plot level (e.g. Antón-Fernández and Astrup 2012; Melo et al. 2017), sometimes
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discriminating between different harvest modalities (e.g. Sterba et al. 2000). Fortin (2014) fitted
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tree-level models using data from plots located in areas with ongoing thinning operations.
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A more informative method would be the combination of both levels, allowing for thinned
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and unthinned plots in the training dataset. This implies two independent logistic models, with
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potentially different covariates in each one of them. These combined models would predict the
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harvest probabilities for a particular plot and for a particular tree within this plot given that it
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will be harvested. This two-level alternative was developed in the context of forest dynamics
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simulators by Thurnher et al. (2011). A similar approach was used by Eastaugh and Hasenauer
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(2012), who substituted predictions of the proportion of harvested stand volume in thinned plots
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for tree-level harvest probabilities.
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Given the hierarchical structure of most forest inventories, correlations between observa-
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tions from the same plot or measurement year are likely to occur. The consequence of not
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dealing with these correlations is the underestimation of the variance of the fixed-effects param-
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eter estimates (Gregoire et al. 1995). The correlation issue has been addressed through copulas
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in tree-level harvest models (Fortin et al. 2013; Delisle-Boulianne et al. 2014). Mixed-effects
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modeling is an alternative way to cope with correlations and is the most popular in forest sci-
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ence. Standard statistical software such as SAS or R allows for the inclusion of random effects
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in logistic models and could be easily used to improve the existing two-level harvest models
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as well. However, a major challenge remains: the covariance between the random effects in-
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cluded in both models cannot be ascertained. This could only be achieved by simultaneously
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estimating the parameters from both models.
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One way to do this is to define the joint distribution of the probability of a plot-level harvest
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and the probabilities of individual-tree harvests conditional on harvest occurrence at plot level.
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Depicting this joint probability was the main objective of the present study. The approach was
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inspired by the mathematical description of the so-called zero-inflated distributions, i.e distri-
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butions holding more zeros than those expected at random (cf. Lambert 1992). Zero-inflated
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distributions are basically defined through the coupling of a Bernoulli distribution, which yields
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the probability of observing a zero event, with another distribution that yields the probability
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of observing non-zero events. This can also be interpreted as an event involving two processes
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where one is conditional to the other. In the current context, harvesting a particular plot can be
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seen as a first Bernoulli process, while harvesting a tree within this plot can be interpreted as
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another Bernoulli process conditional on the first one.
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In order to illustrate the approach, we chose the case study of even-aged mixed stands of
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oak (Quercus spp.) and beech (Fagus sylvatica L.) in Northern France. Specifically, we used
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a network of permanent plots that was set up to investigate different thinning schemes for both
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species. A two-level harvest model with mixed effects was fitted to these data and random
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effects at the plot and tree levels were allowed to be correlated.
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2.1
MATERIAL AND METHODS
DATA
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The data used in the present study were selected from a series of historical and ongoing sil-
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vicultural experiments monitored by the Laboratoire d’Etude de Ressources Forêt-Bois (LER-
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FoB) and that were set up between 1883 and 1956. This compilation constitutes the LERFoB
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permanent-plot network, which is primarily aimed at analysing the effect of management on
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growth and mortality patterns in even-aged sessile oak (Quercus petraea (Matt.) Liebl.) and
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European beech stands across Northern France. In this respect, low thinnings were applied with
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the objective of keeping the stand density below a given value. All control plots (i.e those never
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thinned) were discarded for the purposes of this study. Given the heterogeneous data origin,
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diameter distribution, plot size and measurement intervals notably changed from one site to an-
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other. This information is provided in Table 1 together with the proportion of occasions where
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thinnings took place.
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(Insert Table 1 here)
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The data we used comprised over 175 000 records of diameter at breast height (DBH, 1.3 m)
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from some 33 000 trees within 75 plots in Northern France (Fig. 1) measured between 1922
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and 2012. For each DBH measurement tree status was also recorded (alive, dead or recently
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harvested; see Table 2 for a summary of harvested trees’ proportions). While beech and/or oak
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predominate, other species were also present of which European hornbeam (Carpinus betulus
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L.) was the most common among them.
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(Insert Fig. 1)
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(Insert Table 2 here)
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2.2
S TATISTICAL DEVELOPMENTS
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The probability that a harvest takes place in a plot and that some trees in this plot are then
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harvested can be thought of as a joint probability distribution derived from many univariate
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Bernoulli distributions. Let us assign indices i, j and k to the plot, the tree and the measure-
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ment year, respectively. Plot-level harvest occurrence can be seen as a Bernoulli process with
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outcome qik = 1 and Pr(qik = 1) = λik if plot i is harvested at year k. Otherwise, qik = 0 and
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Pr(qik = 0) = 1 − λik . The probability mass function f (qik ; λik ) of plot-level harvest occurrence
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is then
q
f (qik ; λik ) = λikik (1 − λik )1−qik 107
(1)
Similarly, the harvest of individual trees is also a Bernoulli process with outcome ui jk = 1
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when tree j is felled down in plot i at year k, or ui jk = 0 otherwise. As the harvest of a tree
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is conditional on the harvest occurrence at plot level, the probability that ui jk = 1 needs to be
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accordingly defined as Pr(ui jk = 1 | qik = 1) = πi jk . Likewise, the probability of a tree being
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left standing after a thinning operation is Pr(ui jk = 0 | qik = 1) = 1 − πi jk . The joint probability
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mass function of observing qik and u i·k = (ui1k , ui2k , . . . , uinik k )T conditional on qik = 1 is then u
Pr(qik , u i·k ) = (1 − λik )1−qik λik
nik πi jki jk (1 − πi jk )1−ui jk ∏ j=1
!qik
ik 1 − ∏nj=1 (1 − πi jk )
(2)
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ik The denominator 1 − ∏nj=1 (1 − πi jk ) in Eq. 2 is the probability that at least one tree is
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harvested. This truncation in zero is required since a plot-level harvest occurrence cannot result
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in no harvested tree by definition.
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If parameters λik and πi jk are modeled as functions of some covariates, then it is possible to
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link the harvest probabilities to management, site conditions or other factors thought to influence
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the occurrence of harvest at plot and tree levels. In order to link λik and πi jk to a set of covariates
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(xxik , z i jk ), we used the logit link function (McCullagh and Nelder 1989), so that
ex ik β 1 + ex ik β ez i jk γ πi jk = 1 + ez i jk γ λik =
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(3) (4)
where β and γ are vectors of parameters. In the end, a model likelihood can be expressed on the basis of Eq. 2:
β , γ | q , u , X, Z) = ∏ ∏ Pr(qik , u i·k | β , γ , x ik , Zi·k )) L(β i
(5)
k
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where matrix Zi·k has its rows equal to the z i jk .
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Due to the hierarchical structure of data the observations within the same plot and year
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may not be independent. Random effects can be specified in the model in order to relax the
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assumption of independence. In order to simplify the derivation, we will focus on plot random
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effects only, but the developments can be extended to year random effects as well.
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Random effects were included in Eqs. 3 and 4 so that
λik (bi,1 ) =
ex ik β +bi,1
1 + ex ik β +bi,1 ez i jk γ +bi,2 πi jk (bi,2 ) = 1 + ez i jk γ +bi,2 128
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(6) (7)
where b i = (bi,1 , bi,2 )T is a vector of plot random effects, which is assumed to follow a bivariate normal distribution, such that b i ∼ N2 (00, G).
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Adapting the Eq. 5 accordingly leads to a conditional likelihood that depends on the unob-
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served random effects b i . Parameter estimation relies on this conditional likelihood marginal-
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ized over the distribution of the random effects (Pinheiro and Bates 2000, p.62)
β , γ , G, | q , u , X, Z) = ∏ L(β i
Z
∏ Pr(qik , u i·k | β , γ , x ik , Zi·k , b i) pdf(bbi, G)dbbi
(8)
k
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where pdf(bbi , G) is the density of the bivariate normal distribution with mean 0 and variance-
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covariance G. This complex likelihood can be maximized using the PROC NLMIXED proce-
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dure available in SAS (SAS Institute Inc. 2008). An example of the code we used in this study
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is shown in Appendix A.
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Matrix G contains the variances of bi,1 and bi,2 on its diagonal whereas the off-diagonal elements consist of the covariance between bi,1 and bi,2 : G=
σ12 σ1,2
σ1,2 σ22
(9)
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It can be reasonably assumed that the covariance between random effects bi,1 and bi,2 is
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non null, which means that the harvest probabilities at plot and tree levels would be somehow
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correlated. This covariance parameter σ1,2 can be estimated together with the rest of parameters
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through the maximization of likelihood in Eq. 8. In contrast, if we assume that σ1,2 = 0, then the
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model would be equivalent to fitting a plot-level and a tree-level model independently from each
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other. We will refer to this last model as the constrained model as opposed to the unconstrained
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model in which the covariance is allowed to be non null.
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2.3
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M ODEL SPECIFICATIONS AND EVALUATION
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Different variables were tested sequentially in vectors x ik and z i jk . Because our dataset in-
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cluded mixed stands, a species effect was specified in interaction with the other covariates. This
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species effect had four classes: “oak”, “beech”, “hornbeam” while all other marginal species
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were grouped into a class called “others”.
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Preliminary fits were carried out on a model without random effects and a visual check of
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the Pearson residuals (Hosmer and Lemeshow 2000, p.155) at the plot and the tree level served
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to assess a suitable combination of variables in x ik and z i jk . These preliminary tests showed
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that πi jk exhibited a complex pattern with respect to the relative DBHi jk , which was calculated
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as rDBHi jk = DBHi jk /MQDik , where MQDik is the mean quadratic diameter of plot i at year
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k. A segmented regression was then used to accommodate this pattern. Two segments were
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required, with rDBHi jk = 0.9 being set as the heuristic joint between them. At the plot level the
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time elapsed since the last harvest (LASTCUTik ) seemed to be the covariate that better predict
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the probability of harvest. For the early measurements the time since the last harvest was either
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unobserved because the plots had never been harvested or the last harvest was carried out before
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the beginning of the monitoring. In these cases, plot basal area (BAik ) was used instead.
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The preliminary model structure was
x ik β = β0 + β1 BAik rik + β2 LASTCUTik (1 − rik )
(10)
z i jk γ = γ0 + (γ1,s + γ2,s mi jk )di jk + (γ3 + γ4 (1 − mi jk ))di2jk
(11)
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where ri jk is a dummy variable that takes the value 1 if the time elapsed since the last harvest
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is unknown or 0 otherwise; di jk = rDBHi jk − 0.9 is a transformation that allows both segments
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to converge to the same value at the joint; mi jk is another dummy variable that equals 1 if
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di jk ≥ 0, and 0 otherwise; s stands for the species index.
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The aforementioned Pearson residuals were computed over regular intervals of a particular
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covariate to identify any lack of fit. For each interval g of this covariate, we defined a subset
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of observed events as well as a their corresponding predicted probabilities. If y¯g and y¯ˆg are
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the means of the observations and the predictions, respectively, then the Pearson residual for
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interval g (resg ) can be calculated as y¯g − y¯ˆg resg = q y¯ˆg (1 − y¯ˆg )/ng
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(12)
where ng is the number of observations in interval g. As a result of the joint probability used in Eq. 2, the observations were qik · ui jk whereas the predictions were calculated as λˆ ik · πˆi jk .
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Once the covariates were specified the model was re-fitted with plot and year random ef-
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fects. Two fits were carried out: a first one where random effects were allowed to correlate
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(unconstrained model) and a second one with independent random effects (constrained model).
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A major issue with the predictions of this kind of mixed-effects models is that the predictions
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with the random effects set to 0 are not population-averaged predictions (McCulloch et al. 2008,
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p.190). One method for obtaining population-averaged predictions is to integrate the predictions
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which are conditional on the random effects over the distribution of the random effects, just like
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what is done for the likelihood function 8. This integral has no closed-form solution and can be
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easily approximated using Gauss-Hermite quadrature (Fortin 2013).
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Pearson residuals were computed based on approximated population-averaged predictions
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and their pattern with respect to the different covariates were checked again. The goodness-of-
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fit of the resulting models was assessed using Akaike’s Information Criterion (AIC).
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The unconstrained model was also evaluated based on the Hosmer-Lemeshow test (Hosmer
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and Lemeshow 2000). The test relies on a statistic that is computed similarly to the aforemen-
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tioned Pearson residuals. Predictions of the joint probability that a plot be harvested and that
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a particular tree in this plot be harvested were ranked and grouped according to predictions
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deciles. For each group the squared differences between the mean value of predictions and ob-
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servations were standardized and summed. The Hosmer-Lemeshow statistic is asymptotically
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distributed as a χ 2 with 8 degrees of freedom under the null hypothesis that observed values
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do not significantly differ from the predictions (Hosmer and Lemeshow 2000). All predictions
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were pseudo-independent and population-averaged. This was achieved through a 10-fold cross-
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validation and the marginalisation of the resulting predictions.
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RESULTS
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When simultaneously tested in both parts of the unconstrained model only the year random
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effects improved the model fit with an AIC of 98 345 compared with 103 041 for a model with-
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out random effects. The resulting parameter estimates are shown in Table 3. The unconstrained
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model proved only slightly better than the constrained one (AIC 98 353).
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(Insert Table 3 here)
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Although the Pearson residuals did not show any evidence of lack of fit when the uncon-
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strained model was being fitted, the Hosmer-Lemeshow statistic revealed a significant lack of
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agreement between observations and predictions (χ82 = 1963.99; p-value < 0.0001). Graphi-
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cal comparison of ranked observations and predictions suggested that this divergence could be
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related to an underestimation of the events within the 10th decile (Fig. 2). Concordance was
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evident for all other groups.
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(Insert Fig. 2 here)
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Based on the estimated elements of matrix G, the correlation between the random effects q c2 σ c2 . In our case study, a positive correlation of could be roughly estimated as ρˆ = σˆ 1,2 / σ 1 2
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0.66 was found. This means that if the probability that a plot is harvested is higher than the
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average of the population for a given year, the individual-tree probabilities would also be larger
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than those of the population.
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In Figs. 3 and 4 we graphically illustrate how the different covariates affect the probabilities
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that a plot or a tree is harvested. The ranges of the covariates were set according to the values
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found in the dataset. At the tree level, we chose two different types of plots, a plot in a young
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stand and a plot in a mature stand. The mean quadratic diameter in these two plots was assumed
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to be 20 cm and 50 cm for the young and the mature stands, respectively.
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(Insert Fig. 3 here)
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The plot-level harvest probability increased from 0.45 to 0.70 as the time since the last har-
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vest increased from 1 to 10 years (Fig. 3a). When the date of the last harvest was unknown
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the predicted probability smoothly increased along with plot basal area and the predicted prob-
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abilities ranged from approximately 0.6 to 0.7 for basal areas of 18 to 40 m2 ha−1 , respectively
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(Fig. 3b).
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The effect of rDBH on the tree-level harvest probabilities followed a similar pattern in what
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we defined as a mature and a young stand (Fig. 4). Oak trees with small rDBH, i.e. when
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DBH is smaller than the mean quadratic diameter, were more intensely harvested than beech
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trees. For a rDBH of 0.50, the predicted probabilities of oak were approximately twice those of
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beech in both stands. Larger trees of both species were seldom harvested with probabilities of
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0.10 when the rDBH was close to 1.5. It was only when DBH was twice as large as plot mean
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quadratic diameter, that the predicted probabilities increased for beech trees, with values close
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to 0.25. In contrast, oak individuals did not seem to be harvested in practice in these situations.
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Such large trees with rDBH close to 2 only occurred in the young stand.
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(Insert Fig. 4 here)
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DISCUSSION
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The present study sets the grounds for simultaneous parameter estimation in two-level har-
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vest models. The statistical developments needed for this implementation led to a relevant
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progress in the current state-of-the-art of zero-inflated models. However, the main achievement
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of our approach stems from the fact that the full variance-covariance matrix of the random ef-
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fects can be estimated when random effects are present in both the plot-level and the tree-level
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parts of the model.
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Using zero-inflated mixed models with correlated random effects is not new in forestry.
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Calama et al. (2011) fitted a zero-inflated log-normal model to forecast the weight of cones
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produced by a pine species whereas Manso et al. (2014) worked with a zero-inflated bino-
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mial model to predict seed predation and in both cases the random effect correlation was taken
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into account. In contrast, two-level harvest models had never been analysed under a zero-
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inflated/altered framework. As a consequence a simultaneous parameter estimation was not
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possible and in turn the correlation between the random effects could not be considered. The
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fact that the likelihood formulation of two-level harvest models with mixed effects is more
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complicated than those of the previous examples may have prevented modelers from testing
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this approach.
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In forestry zero-inflated and zero-altered models have been aimed at modeling various re-
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sponses such as recruitment (Fortin and DeBlois 2007; Ledo et al. 2015), seed predation (Manso
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et al. 2014), cone production (Calama et al. 2011), wildfire damage (Guo et al. 2016) or LiDAR-
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based tree detection (Korhonen et al. 2016). One application that falls conceptually closer to
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harvesting is that of tree mortality modeling. In this respect some authors have proposed the
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Poisson distribution as the underlying process driving mortality in a given plot providing that
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mortality takes place at all so that the number of dead trees in such a context can be represented
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as a count variable (Affleck 2006).
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A weakness of the Poisson distribution is that it is has no upper bound whereas there is
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only a finite number of individuals that can die in a given plot. In order to deal with this issue
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an offset variable, that is also called the exposure, can be specified (McCullagh and Nelder
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1989, p.206). As a result, model predictions can be interpreted as a proportion. A more natural
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approach would be to treat the conditional process as a binary response as well, which would
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lead to a zero-inflated/altered binomial (e.g. Hall 2000)). The binomial distribution therefore
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provides the expectation that a tree dies, providing that all trees share the same probability.
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The same rationale could apply to achieve simultaneous parameter estimation in two-level
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harvest models as follows: given that a plot is going to be harvested, the number of harvested
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trees could be assumed Poisson- or binomial-distributed. A major limitation of either approach
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is that the same probability is assumed for every single tree, as mentioned before. This assump-
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tion is not valid as shown in previous works on the harvesting of individual trees (Thurnher
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et al. 2011; Fortin et al. 2013; Delisle-Boulianne et al. 2014; Fortin 2014). In partial harvesting
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each tree has a distinctive probability of being harvested that depends on a set of covariates.
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The Poisson approach does not allow for an individual probability assignment whilst the bino-
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mial distribution can be adapted. Given that a binomial process is simply the repetition of a
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Bernoulli trial, the binomial distribution that represents the harvesting of the individual trees
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can be derived from the individual probabilities. This concept has been defined as a binomial
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distribution with unequal probabilities (see Friedman 1984).
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The distinction between zero-inflated and zero-altered models lies in the truncation of zero
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in the second distribution that composes the joint distribution (Zeileis et al. 2008). This trunca-
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tion implies that the zeros cannot be observed if a harvest is carried out. This was true in our
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case study because we modeled the plot-level occurrence and the plots were large in area. If the
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plots are small and the focus is on the stand-level occurrence, it may happen that no trees are
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harvested in a given plot simply because it was too small. In such a case a pure zero-inflated
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formulation would be more appropriate.
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The sign of the correlation between the random effects of the plot-level and the tree-level
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parts in a zero-inflated model like that applied in our case study can be interpreted. A positive
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correlation implies that the random effects tend to be both positive or negative. If the random
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effects are positive then the plot has a higher probability of being harvested and if it is, more
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trees will be harvested within that plot. Negative random effects indicate that a plot has a lower
291
probability of being harvested and even if it is, fewer trees will be harvested. In practice, the
292
main implication of a positive correlation is that the deviation of predictions with respect to
293
the mean would be larger than what is predicted when the random effects are uncorrelated.
294
Under the assumption of independent random effects, probabilities higher than the mean at the
295
plot level may be compensated by lower probabilities at the tree level and vice versa, which
296
would eventually lead to underestimating the true variance of the prediction errors. It is also
297
well known that neglecting random effects leads to an underestimation of the variance of the
298
estimated of the fixed effects Gregoire et al. (1995), which may result in an additional impact
299
on the uncertainty of predictions.
300
The method presented in this study was developed to account for the effects of correla-
301
tions between random effects in uncertainty assessment, but it was not possible to quantify the
P REDICTION OF PLOT- LEVEL AND TREE - LEVEL HARVEST OCCURRENCES
14
302
implications of ignoring them. The unconstrained model did not markedly outperformed the
303
constrained one in terms of AIC, making further comparisons via simulation exercises redun-
304
dant. Nevertheless, our approach makes it possible to test this possibility and to compare the
305
model with simpler alternatives, which may prove useful in other cases.
306
Concerning covariates, the variable with the most dominant effect on the probability of
307
harvest at the plot level was the time elapsed since the last harvest; however, this variable may
308
be unknown in cases where harvesting has not yet occurred or took place prior to monitoring.
309
Such gaps in data can be resolved by using dummy variables and we managed to use the plot
310
basal area in place of the time since the last harvest when the latter was unknown. In preliminary
311
trials we assumed that basal area would be the major driver of harvest occurrence and that the
312
time elapsed since the last harvest would not be needed, however the preliminary model resulted
313
in a poor fit. The significant effect of the time-since-last-harvest on predictions of plot-harvest-
314
occurence can be explained by the management of these stands. Once the stem exclusion stage
315
as defined by Oliver and Larson (1996) is over, these stands are usually thinned on a more or
316
less regular basis and the density is not a major criterion in the management (see ONF 2007;
317
Sardin 2008).
318
At the tree level the relative diameter and species group were the two variables that were
319
found to have a significant effect on the harvest occurrence. Tree diameter and species were also
320
the most important covariates used in other studies (Fortin 2014; Thurnher et al. 2011). In this
321
respect, we found lower probabilities for larger stems in contrast to the findings of other studies
322
(e.g. Fortin 2014). Management differences may explain this divergence. In Fortin (2014) the
323
stands were mainly uneven-aged and managed using selection cutting while in our case the
324
stands are even-aged and low thinnings are applied. Therefore it is not surprising to find higher
325
probabilities of harvest in smaller stems (Fig. 4). The different species-specific patterns found
326
in the present study are related to species’ ecology. For instance, the fact that oak is harvested
327
more than beech is related to its tolerance to shade; beech is very shade-tolerant whereas oak is
328
intermediate. Suppressed oak trees have a low probability of survival which has been evidenced
329
in Manso et al. (2015). In this context, forest managers prefer to harvest them while they are
330
still alive and valuable.
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5
331
15
CONCLUSIONS
332
The covariance of random effects between plot-level and tree-level harvest models cannot
333
be estimated if the models are fitted independently. In the present study an approach inspired by
334
zero-inflated modeling was developed which simultaneously estimates parameters at both plot
335
and tree levels and subsequently the covariance of random effects. The method was applied in
336
a case study and compared to a classical approach of independently estimating parameters. The
337
method presented did not clearly outperform the existing approach; however we believe that it
338
is still a relevant contribution to harvest modeling, given the potential implications of neglecting
339
random effect covariance on uncertainty assessment.
340
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Affleck, D. L. (2006). Poisson mixture models for regression analysis of stand-level mortality.
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Antón-Fernández, C. and Astrup, R. (2012). Empirical harvest models and their use in regional
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business-as-usual scenarios of timber supply and carbon stock development. Scandinavian
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Calama, R., Mutke, S., Tomé, J., Gordo, J., Montero, G., and Tomé, M. (2011). Modelling
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spatial and temporal variability in a zero-inflated variable: The case of stone pine (Pinus
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pinea L.) cone production. Ecological Modelling, 222(3):606–618.
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Delisle-Boulianne, S., Fortin, M., Achim, A., and Pothier, D. (2014). Modelling stem selection
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in northern hardwood stands: assessing the effects of tree vigour and spatial correlations
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using a copula approach. Forestry, 87:607–617.
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Eastaugh, C. S. and Hasenauer, H. (2012). A statistical thinning model for initialising largescale ecosystem models. Scandinavian Journal of Forest Research, 27(6):567–577.
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Fortin, M. (2013). Population-averaged predictions with generalized linear mixed-effects mod-
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els in forestry: an estimator based on Gauss-Hermite quadrature. Canadian Journal of Forest
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Research, 43:129–138.
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Fortin, M. (2014). Using a segmented logistic model to predict trees to be harvested in forest
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growth forecasts. Forest Systems, 23(1):139.
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Fortin, M. and DeBlois, J. (2007). Modeling tree recruitment with zero-inflated models: The
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example of hardwood stands in southern Quebec, Canada. Forest Science, 53(4):529–539.
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Fortin, M., Delisle-Boulianne, S., and Pothier, D. (2013). Considering spatial correlations
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between binary response variables in forestry: an example applied to tree harvest modeling.
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Forestry, 59(3):253–260.
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Friedman, M. F. (1984). On the extended binomial distribution. Computers & Operations Research, 11(3):241–243.
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Gregoire, T., Schabenberger, O., and Barrett, J. (1995). Linear modelling of irregularly spaced,
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unbalanced, longitudinal data from permanent-plot measurements. Canadian Journal of For-
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Guo, F., Wang, G., Innes, J., Ma, Z., Liu, A., and Lin, Y. (2016). Comparison of six generalized
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China. Journal of Forestry Research, 27(2):379–388.
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Hall, D. (2000). Zero-inflated poisson and binomial regression with random effects: a case study. Biometrics, 56:1030–1039. Hosmer, D. J. and Lemeshow, S. (2000). Applied logistic regression. John Wiley & Sons, New York, 2nd edition.
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Korhonen, L., Salas, C., Østgård, T., Lien, V., Gobakken, T., and Næsset, E. (2016). Predicting
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the occurrence of large-diameter trees using airborne laser scanning. Canadian Journal of
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Lambert, D. (1992). Zero-inflated poisson regression, with an application to defects in manufacturing. Technometrics, 34(1):1–14.
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Ledo, A., Cayuela, L., Manso, R., and Condés, S. (2015). Recruitment of woody plants in a
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cloud forest: a combination of spatial mechanisms. Journal of Vegetation Science, 26(5):876–
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888.
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Manso, R., Morneau, F., Ningre, F., and Fortin, M. (2015). Incorporating stochasticity from
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extreme climatic events and multi-species competition relationships into single-tree mortality
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models. Forest Ecology and Management, 354:243–253. Manso, R., Pardos, M., and Calama, R. (2014). Climatic factors control rodent seed predation in Pinus pinea L. stands in Central Spain. Annals of Forest Science, 71(8):873–883. McCullagh, P. and Nelder, J. A. (1989). Generalized linear models. Monographs of Statistics and Applied Probability 37. Chapman&Hall, New York, 2 edition. McCulloch, C., Searle, S., and Neuhaus, J. M. (2008). Generalized, linear, and mixed models. John Wiley & Sons, New York.
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Melo, L. C., Schneider, R., Manso, R., Saucier, J.-P., and Fortin, M. (2017). Using survival
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analysis to predict the harvesting of forest stands in Quebec, Canada. Canadian Journal of
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Oliver, C. D. and Larson, B. C. (1996). Forest Stand Dynamics. Updated Edition. John Wiley and Sons, New York, USA. ONF (2007). Gestion des hêtraies dans les forêts publiques françaises. Office National des Forêts.
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Pinheiro, J. and Bates, D. (2000). Mixed effects models in S and S-PLUS. Springer, New York.
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Pukkala, T. and Miina, J. (1998). Tree-selection algorithms for optimizing thinning using a
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distance-dependent growth model. Canadian Journal of Forest Research, 28:693–702.
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Sardin, T. (2008). Chênaies continentales. Guide des sylvicultures. Office National des Forêts.
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SAS Institute Inc. (2008). SAS/STAT 9.2 User’s Guide. SAS Institute Inc., Cary, NC.
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Sterba, H., Golser, M., Moser, M., and Schadauer, K. (2000). A timber harvesting model for
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Austria. Computers and Electronics in Agriculture, 28(2):133–149. Thurnher, C., Klopf, M., and Hasenauer, H. (2011). Forests in transition: a harvesting model for uneven-aged mixed species forests in Austria. Forestry, 84(5):517–526. Zeileis, A., Kleiber, C., and Jackman, S. (2008). Regression models for count data in R. Journal of Statistical Software, 27(8).
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A
18
IMPLEMENTATION OF THE MODEL IN SAS
412
The next code reproduces the necessary elements to fit the two-level harvest model intro-
413
duced in the present paper under the unconstrained assumption for the variance-covariance ma-
414
trix of the random effects.
415
/*******************************************************************/
416
proc nlmixed data=... ;
417
parms
418
g0 = , ... , g3 = /* Initial values of parameters, plot-level */
419
b0 = , ... , b10 = /* Initial values of parameters, tree-level */
420
su2 = , sv2 = , suv = ; /* Initial values variance random effects */
421
/* arrays for the different covariates. 1433 is the maximum number of
422
observations in a plot in the present example */
423
array ssd [1433] ssd1-ssd1433; /* scaled mean square diameter */
424
array het [1433] bee1-bee1433; /* dummy species = beech */
425
array chx [1433] oak1-oak1433; /* dummy species = oak */
426
array cha [1433] hor1-hor1433; /* dummy species = hornbean */
427
array aut [1433] oth1-oth1433; /* dummy species = others */
428
array threshold [1433] thr1-thr1433; /* dummy dbh > threshold */
429
array cut[1433] cut1-cut1433; /* dummy cut tree */
430
allcutProb = 1;
431
likTreeLevel = 0;
432
do i = 1 to N; /* N is the number of trees in each plot */
433
marginalLinearTermsTree =
b0 +
434
(b1*bee[i] + b2*oak[i] + b3*hor[i] + b4*oth[i] +
435
(b5*bee[i] + b6*oak[i] + b7*hor[i] + b8*oth[i])*thr[i])*ssd[i] +
436
(b9 + 10*thr[i])*(ssd[i]**2);
437
linearTermsTree =
marginalLinearTermsTree + u;
438
pi = exp(linearTermsTree)/(1 + exp(linearTermsTree));
439
allcutProb = allcutProb*(1-pi);
440
likTreeLevel = likTreeLevel + cut[i]*log(pi) + (1-cut[i])*log(1-pi);
P REDICTION OF PLOT- LEVEL AND TREE - LEVEL HARVEST OCCURRENCES
441
end;
442
marginalLinearTermsPlot = notCutYet*(g0 + g1*ST) +
19
443
alreadyCut*(g2 + g3*timeSinceLastCut);
444
linearTermsPlot = marginalLinearTermsPlot + v;
445
p = exp(linearTermsPlot)/(1 + exp(linearTermsPlot));
446
logLik = (1-cutPlot)*log(1 - p) + cutPlot*log(p) + cutPlot*likTreeLevel cutPlot*log(1 - allcutProb); /* cutPlot: dummy thinned plot */
447
448
model cutPlot $\sim$ general(logLik);
449
random u v $\sim$ normal([0, 0], [su2, suv, sv2]) subject= year;
450
run;
451
/*******************************************************************/
Table
1
Tables
1
2
Experimental site All´ ee de Blois Bois Brochet Beaulieu Butte de Tir Camp Souverain Camp Cusson Charlemagne Chatellier Chavigny Chr´ etiennette Cl´ es des Foss´ es Cˆ otes aux Hˆ etreaux Croix de Saverne Ducellier Epic´ eas Faˆıte Grande Bouzule Grand Pierrier Hallet Hermousset Lacharmaie Fontaine aux Ordons Launay-Morel M. des Cordeliers Morat Mortefert Pauverts Plantonn´ ee Plˆ o du Poteau Pr´ e des Seigneurs Puiseux-en-Retz Rennweg Carr´ e latin de R´ eno Richebou rg Sablonni` eres Sainte Marie Tr´ esor” Verbamont
Initial date 1927 1931 1923 1959 1927 1931 1923 1934 1923 1922 1931 1927 1946 1945 1923 1922 1928 1923 1951 1934 1925 1923 1945 1928 1931 1922 1928 1981 1931 1922 1922 1951 1989 1931 1966 1958 1959 1923
Observations 2946 6577 3032 5821 769 14501 2606 5808 2209 2366 2326 1260 945 1909 3260 12671 19860 1128 13575 9657 589 1797 330 9024 798 569 14743 589 639 1167 3035 1388 10354 2100 8521 1310 5478 766
DBH (cm) 20.4-85.3 10.5-85 9.5-50.6 7.6-53.2 25.1-85.3 7.3-90.7 6.4-80.5 7-85.3 6.4-68.8 7-73.8 14.3-96.1 19.7-87.5 6.7-58.3 15.6-75.8 9.2-75.1 4.5-61.8 6-70.3 8-66.2 6-65.9 5.7-81.8 35.7-77.0 13.4-80.2 30.9-144.2 14.6-87.2 14-106 39.2-106 6-74.2 6-43.6 25.5-76.7 29.6-95.5 6.4-78.9 9.9-44.2 5.7-54.7 17.5-112.7 7.0-65.3 3.8-41.7 6-69.4 17.8-88.8
Basal area (m2 ha−1 ) 23.8-37.4 20.8-29.7 21.2-27.2 19.6-34.7 25.9-36.6 20.0-43.1 29.7-47.6 22.2-37.2 31.9-45.2 18.2-39.6 30.4-38.7 30.2-37 28.6-30.5 16.6-26.1 21-44.2 10.1-33.6 19.1-36.7 31.2-39 15.3-35.5 19.8-34.6 2.0-23.6 27.9-33.4 6.8-28.1 21.8-35.8 15.6-46.3 15.6-31.6 20.1-33.9 4.3-23.4 5.8-21.5 34.7-44.4 21-31.3 25.5-29.1 16.3-34.1 30.0-34.6 10.9-35.4 22.6-23.5 23.6-31.9 9.2-21.8
Density (stems ha−1 ) 145-304 94-602 352-673 168-1102 138-217 81-813 156-988 96-611 155-1115 96-531 98-483 163-275 277-335 91-283 160-776 165-2155 147-1223 156-1068 205-1162 122-1433 7-126 145-309 20-89 92-567 34-137 35-107 119-1242 84-376 28-156 107-203 94-1572 684-728 295-1010 88-212 81863 606-740 199-878 58-172 Interval length (years) 1-10 1-10 2-8 1-9 1-5 1-7 2-10 3-8 2-10 3-10 1-10 2-9 5 1-10 2-10 2-10 1-9 1-10 2-10 3-10 2-6 2-8 4-6 1-9 5-7 3-7 1-10 1-10 1-6 3-10 3-10 5 1-5 2-8 1-10 4 4-10 1-8
Plot size (ha) 1 1 1 0.5 1 1 0.2 1 0.2 0.3 1 1 1 1 0.2 0.2 1 0.2 1 1 2 1 2 1 2 1 1 0.5 0.9 1 0.5 0.5 0.2 2 0.8 1 1 1
Prop. intervals with cuts 0.57 0.67 0.67 0.5 0.4 0.61 0.82 0.6 0.62 0.62 0.73 0.83 0.33 0.7 0.53 0.71 0.58 0.7 0.79 0.89 0.8 0.75 1 0.67 0.8 0.57 0.58 0.4 0.4 0.57 0.7 0.75 0.62 0.71 0.62 1 1 0.5
Table 1: Summary of the dataset by experiment including the range of the main variables for the considered measurement periods.
Table 2: Frequencies of the proportion of cut trees (all, beech, oak) in the years and plots where thinnigs took place. Proportions are set over the total and per species (when present in the plot). proportion 0-0.05 0.05-0.15 0.15-0.25 0.25-0.35 0.35-0.45 0.45-0.55 0.55-0.65 0.65-0.75 0.75-0.85 0.85-0.95 0.95-1
all species/total 93 119 138 62 20 3 3 1 1 0 1
beech/total beech 50 74 78 47 12 7 3 6 2 2 12
3
beech/total 121 72 59 27 5 3 2 0 0 0 4
oak/total oak 45 66 99 39 16 6 3 3 3 2 16
oak/total 108 71 77 29 7 1 1 0 1 0 1
Table 3: Maximum likelihood parameters estimates and standard errors of the simulta2 2 neous plot and tree thinning model. σyear,b and σyear,b stand for the variance of the year 1 2 random effects for the plot and tree models, while σyear,b1 ,b2 is the covariance between them. Parameter β0 β1 β2 γ0 γ1,beech γ1,oak γ1,hornbeam γ1,others γ2,beech γ2,oak γ2,hornbeam γ2,others γ3 γ4 2 σyear,b 1 2 σyear,b 2 σyear,b1 ,b2
estimate -0.3694 0.0387 0.1760 -2.1306 -4.0513 -5.3613 -0.9210 -2.3803 -1.2513 -3.7132 -2.0291 -0.1231 4.0450 -11.1102 1.5807 1.5076 1.0155
4
standard error 0.2862 0.0135 0.0368 0.1634 0.1587 0.1508 0.5513 0.7612 0.3242 0.3038 0.6480 1.5476 0.1989 0.4614 -
Figure
Figure
Figure
Figure
Figure Captions
1
Figure Captions
2
• Figure 1. Location of sample plots
3
• Figure 2. Predicted (circles) and observed (triangles) numbers of events by decile
4
in the Hosmer-Lemeshow test
5
• Effect of the time since the last harvest (a) or the basal area (b) when this time is
6
unknown on the plot-level probabilities of harvest. The 0.95 confidence envelopes
7
are represented in gray
8
• Figure 4. Effect of the relative DBH, i.e the ratio between tree dbh and plot
9
mean quadratic diameter, on the conditional probabilities that a tree is harvested
10
in a mature stand (a) and a young stand (b). The 0.95 confidence envelopes are
11
represented in gray
1
